Chapter 2: First Order DE 2.4 Linear vs. Nonlinear DEs
|
|
- Katherine Ray
- 5 years ago
- Views:
Transcription
1 Chapter 2: First Order DE 2.4 Linear vs. Nonlinear DEs
2 First Order DE 2.4 Linear vs. Nonlinear DE We recall the general form of the First Oreder DEs (FODE): dy = f(t, y) (1) dt where f(t, y) is a function of both the dependent variable t and the (unknown) dependent variable y. A DE, with an initial value condition y(t 0 ) = y 0 is called an Initial Value Problem (IVP). In 2.1, we worked with Linear FODEs. In this section ( 2.4), we compare Linear vs. Nonlinear FODE.
3 Linear vs. Nonlinear DE A 1 st -order DE is called Linear, if it has the form Else, it is called nonlinear. Given a DE (linear or not), we ask: y + p(t)y = g(t) (2) Does all initial value problems (IVP) have a solution y? If there is a solution y = y(t), what is its domain? When an IVP has a solution, is it unique? In other words, could an IVP have more than one solution? For linear IVP, it is easier to answer these questions.
4 Existance and Uniqueness of solutions Theorem 2.4.1: Consider the 1 st -order linear IVP { y + p(t)y = g(t) y(t 0 ) = y 0 (3) Assume p(t), g(t) are continuous on an interval I : α < t < β and t 0 is in I. Then, The IVP (3) has a solution y = ϕ(t). The domain of y = ϕ(t) is I. The solution y = ϕ(t) is unique, on I.
5 Continued 2.4 Linear vs. Nonlinear DE The proof of the existance is done by defining integrating factor µ(t) = exp p(t)dt, as we solved problems in 2.1. For nonlinear FO-IVP y = f(t, y), y(t 0 ) = y 0 : In general, the solutions of such DEs need not be unique. But theorem would be true in a more restricted sense (see theorem 2.4.2).
6 Sample I (Ex 1-6) Sample II (Ex 1-6) Exercise 4, pp 76 Consider the initial value problem (IVP) { (4 t 2 )y + 2ty = 3t 2 y( 3) = 1 Determine the (without solving) an interval in which this IVP has a unique solution. Write the equation in the form (3), as in theorem 2.4.1: { y + 2t 4 t 2 y = 3t2 4 t 2 y( 3) = 1
7 Continued 2.4 Linear vs. Nonlinear DE Sample I (Ex 1-6) Sample II (Ex 1-6) p(t) = 2t 4 t 2 is not defined at t = 2, 2. Split the number line as: (, 2),( 2, 2),(2, ). Both p(t) = 2t, g(t) = 3t2 are continuous on the 4 t 2 4 t 2 intervals (, 2),( 2, 2),(2, ). The initial t-value t = 3 is in (, 2) By theorem the IVP has a unique solution on (, 2).
8 y 2.4 Linear vs. Nonlinear DE Sample I (Ex 1-6) Sample II (Ex 1-6) The Direction Field: There should be a vertical tangent at x = 2, which is not clear form the dfield. y = ( 2 t y + 3 t 2 )/(4 t 2 ) t
9 Sample I (Ex 1-6) Sample II (Ex 1-6) Exercise 3, pp 76 Consider the initial value problem (IVP) { y +(tan t)y = sin t y(π) = 0 Determine the (without solving) an interval in which this IVP has a unique solution. The equation in the form (3), as in theorem tan t is not defined at t = ± π 2, 3π 2, 5π 2,...
10 Continued 2.4 Linear vs. Nonlinear DE Sample I (Ex 1-6) Sample II (Ex 1-6) Accordingly, split the number line as: (, π 2, π ) (, π 2 2, 3π 2 ), Both p(t) = tan t, g(t) = sin t are continuous on these intervals. The initial t-value t = π is in ( π, ) 3π 2 2. By ( theorem the IVP has a unique solution on π, ) 3π 2 2.
11 Nonlinear DE Nonlinear DEs would not behave as nicely as in theorem Uniqueness is not guaranteed. Solutions, if exist, may come out in an implicit form. Some DEs may not have an analytic solution. In such cases, numerical solutions would be an option.
12 Assignments and Homework Read Example 1, 2 ( 2.4) Suggested Problems: Exercise (page 76) Homework: 2.4 See Homework Site!
Homogeneous Equations with Constant Coefficients
Homogeneous Equations with Constant Coefficients MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Spring 2018 General Second Order ODE Second order ODEs have the form
More informationSection 2.1 (First Order) Linear DEs; Method of Integrating Factors. General first order linear DEs Standard Form; y'(t) + p(t) y = g(t)
Section 2.1 (First Order) Linear DEs; Method of Integrating Factors Key Terms/Ideas: General first order linear DEs Standard Form; y'(t) + p(t) y = g(t) Integrating factor; a function μ(t) that transforms
More informationFirst Order Differential Equations Lecture 3
First Order Differential Equations Lecture 3 Dibyajyoti Deb 3.1. Outline of Lecture Differences Between Linear and Nonlinear Equations Exact Equations and Integrating Factors 3.. Differences between Linear
More informationFirst Order ODEs, Part I
Craig J. Sutton craig.j.sutton@dartmouth.edu Department of Mathematics Dartmouth College Math 23 Differential Equations Winter 2013 Outline 1 2 in General 3 The Definition & Technique Example Test for
More informationChapter 4: Higher Order Linear Equations
Chapter 4: Higher Order Linear Equations MATH 351 California State University, Northridge April 7, 2014 MATH 351 (Differential Equations) Ch 4 April 7, 2014 1 / 11 Sec. 4.1: General Theory of nth Order
More informationLinear Independence and the Wronskian
Linear Independence and the Wronskian MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Spring 2018 Operator Notation Let functions p(t) and q(t) be continuous functions
More informationSecond Order Linear Equations
October 13, 2016 1 Second And Higher Order Linear Equations In first part of this chapter, we consider second order linear ordinary linear equations, i.e., a differential equation of the form L[y] = d
More informationy + p(t)y = g(t) for each t I, and that also satisfies the initial condition y(t 0 ) = y 0 where y 0 is an arbitrary prescribed initial value.
p1 Differences Between Linear and Nonlinear Equation Theorem 1: If the function p and g are continuous on an open interval I : α < t < β containing the point t = t 0, then there exists a unique function
More informationFirst-Order ODEs. Chapter Separable Equations. We consider in this chapter differential equations of the form dy (1.1)
Chapter 1 First-Order ODEs We consider in this chapter differential equations of the form dy (1.1) = F (t, y), where F (t, y) is a known smooth function. We wish to solve for y(t). Equation (1.1) is called
More informationChapter 2 Notes, Kohler & Johnson 2e
Contents 2 First Order Differential Equations 2 2.1 First Order Equations - Existence and Uniqueness Theorems......... 2 2.2 Linear First Order Differential Equations.................... 5 2.2.1 First
More informationLecture 2. Classification of Differential Equations and Method of Integrating Factors
Math 245 - Mathematics of Physics and Engineering I Lecture 2. Classification of Differential Equations and Method of Integrating Factors January 11, 2012 Konstantin Zuev (USC) Math 245, Lecture 2 January
More informationFirst Order ODEs, Part II
Craig J. Sutton craig.j.sutton@dartmouth.edu Department of Mathematics Dartmouth College Math 23 Differential Equations Winter 2013 Outline Existence & Uniqueness Theorems 1 Existence & Uniqueness Theorems
More informationLecture Notes in Mathematics. Arkansas Tech University Department of Mathematics
Lecture Notes in Mathematics Arkansas Tech University Department of Mathematics Introductory Notes in Ordinary Differential Equations for Physical Sciences and Engineering Marcel B. Finan c All Rights
More informationLecture Notes for Math 251: ODE and PDE. Lecture 7: 2.4 Differences Between Linear and Nonlinear Equations
Lecture Notes for Math 51: ODE and PDE. Lecture 7:.4 Differences Between Linear and Nonlinear Equations Shawn D. Ryan Spring 01 1 Existence and Uniqueness Last Time: We developed 1st Order ODE models for
More informationMathematical Models. MATH 365 Ordinary Differential Equations. J. Robert Buchanan. Spring Department of Mathematics
Mathematical Models MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Spring 2018 Ordinary Differential Equations The topic of ordinary differential equations (ODEs)
More informationMathematical Models. MATH 365 Ordinary Differential Equations. J. Robert Buchanan. Fall Department of Mathematics
Mathematical Models MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Ordinary Differential Equations The topic of ordinary differential equations (ODEs) is
More informationMATH 353 LECTURE NOTES: WEEK 1 FIRST ORDER ODES
MATH 353 LECTURE NOTES: WEEK 1 FIRST ORDER ODES J. WONG (FALL 2017) What did we cover this week? Basic definitions: DEs, linear operators, homogeneous (linear) ODEs. Solution techniques for some classes
More information2. First Order Linear Equations and Bernoulli s Differential Equation
August 19, 2013 2-1 2. First Order Linear Equations and Bernoulli s Differential Equation First Order Linear Equations A differential equation of the form y + p(t)y = g(t) (1) is called a first order scalar
More informationHW2 Solutions. MATH 20D Fall 2013 Prof: Sun Hui TA: Zezhou Zhang (David) October 14, Checklist: Section 2.6: 1, 3, 6, 8, 10, 15, [20, 22]
HW2 Solutions MATH 20D Fall 2013 Prof: Sun Hui TA: Zezhou Zhang (David) October 14, 2013 Checklist: Section 2.6: 1, 3, 6, 8, 10, 15, [20, 22] Section 3.1: 1, 2, 3, 9, 16, 18, 20, 23 Section 3.2: 1, 2,
More informationLinear Variable coefficient equations (Sect. 1.2) Review: Linear constant coefficient equations
Linear Variable coefficient equations (Sect. 1.2) Review: Linear constant coefficient equations. The Initial Value Problem. Linear variable coefficients equations. The Bernoulli equation: A nonlinear equation.
More informationSolutions to Homework 5, Introduction to Differential Equations, 3450: , Dr. Montero, Spring y 4y = 48t 3.
Solutions to Homework 5, Introduction to Differential Equations, 3450:335-003, Dr. Montero, Spring 2009 Problem 1. Find a particular solution to the differential equation 4y = 48t 3. Solution: First we
More informationMath Homework 3 Solutions. (1 y sin x) dx + (cos x) dy = 0. = sin x =
2.6 #10: Determine if the equation is exact. If so, solve it. Math 315-01 Homework 3 Solutions (1 y sin x) dx + (cos x) dy = 0 Solution: Let P (x, y) = 1 y sin x and Q(x, y) = cos x. Note P = sin x = Q
More informationFirst and Second Order Differential Equations Lecture 4
First and Second Order Differential Equations Lecture 4 Dibyajyoti Deb 4.1. Outline of Lecture The Existence and the Uniqueness Theorem Homogeneous Equations with Constant Coefficients 4.2. The Existence
More informationSMA 208: Ordinary differential equations I
SMA 208: Ordinary differential equations I First Order differential equations Lecturer: Dr. Philip Ngare (Contacts: pngare@uonbi.ac.ke, Tue 12-2 PM) School of Mathematics, University of Nairobi Feb 26,
More informationSection 4.7: Variable-Coefficient Equations
Cauchy-Euler Equations Section 4.7: Variable-Coefficient Equations Before concluding our study of second-order linear DE s, let us summarize what we ve done. In Sections 4.2 and 4.3 we showed how to find
More informationProblem 1 In each of the following problems find the general solution of the given differential
VI Problem 1 dt + 2dy 3y = 0; dt 9dy + 9y = 0. Problem 2 dt + dy 2y = 0, y(0) = 1, y (0) = 1; dt 2 y = 0, y( 2) = 1, y ( 2) = Problem 3 Find the solution of the initial value problem 2 d2 y dt 2 3dy dt
More informationSample Questions, Exam 1 Math 244 Spring 2007
Sample Questions, Exam Math 244 Spring 2007 Remember, on the exam you may use a calculator, but NOT one that can perform symbolic manipulation (remembering derivative and integral formulas are a part of
More informationSecond-Order Linear ODEs
Second-Order Linear ODEs A second order ODE is called linear if it can be written as y + p(t)y + q(t)y = r(t). (0.1) It is called homogeneous if r(t) = 0, and nonhomogeneous otherwise. We shall assume
More informationFind the indicated derivative. 1) Find y(4) if y = 3 sin x. A) y(4) = 3 cos x B) y(4) = 3 sin x C) y(4) = - 3 cos x D) y(4) = - 3 sin x
Assignment 5 Name Find the indicated derivative. ) Find y(4) if y = sin x. ) A) y(4) = cos x B) y(4) = sin x y(4) = - cos x y(4) = - sin x ) y = (csc x + cot x)(csc x - cot x) ) A) y = 0 B) y = y = - csc
More informationSeries Solutions Near an Ordinary Point
Series Solutions Near an Ordinary Point MATH 365 Ordinary Differential Equations J. Robert Buchanan Department of Mathematics Spring 2018 Ordinary Points (1 of 2) Consider the second order linear homogeneous
More informationSolutions of Math 53 Midterm Exam I
Solutions of Math 53 Midterm Exam I Problem 1: (1) [8 points] Draw a direction field for the given differential equation y 0 = t + y. (2) [8 points] Based on the direction field, determine the behavior
More informationSection 2.1 Differential Equation and Solutions
Section 2.1 Differential Equation and Solutions Key Terms: Ordinary Differential Equation (ODE) Independent Variable Order of a DE Partial Differential Equation (PDE) Normal Form Solution General Solution
More informationLinear Variable coefficient equations (Sect. 2.1) Review: Linear constant coefficient equations
Linear Variable coefficient equations (Sect. 2.1) Review: Linear constant coefficient equations. The Initial Value Problem. Linear variable coefficients equations. The Bernoulli equation: A nonlinear equation.
More informationLecture 31. Basic Theory of First Order Linear Systems
Math 245 - Mathematics of Physics and Engineering I Lecture 31. Basic Theory of First Order Linear Systems April 4, 2012 Konstantin Zuev (USC) Math 245, Lecture 31 April 4, 2012 1 / 10 Agenda Existence
More informationWe begin exploring Euler s method by looking at direction fields. Consider the direction field below.
Emma Reid- MA 66, Lesson 9 (SU 17) Euler s Method (.7) So far in this course, we have seen some very special types of first order ODEs. We ve seen methods to solve linear, separable, homogeneous, Bernoulli,
More informationLecture 16. Theory of Second Order Linear Homogeneous ODEs
Math 245 - Mathematics of Physics and Engineering I Lecture 16. Theory of Second Order Linear Homogeneous ODEs February 17, 2012 Konstantin Zuev (USC) Math 245, Lecture 16 February 17, 2012 1 / 12 Agenda
More informationMATH 308 Differential Equations
MATH 308 Differential Equations Summer, 2014, SET 1 JoungDong Kim Set 1: Section 1.1, 1.2, 1.3, 2.1 Chapter 1. Introduction 1. Why do we study Differential Equation? Many of the principles, or laws, underlying
More information2.2 Separable Equations
2.2 Separable Equations Definition A first-order differential equation that can be written in the form Is said to be separable. Note: the variables of a separable equation can be written as Examples Solve
More informationPuzzle 1 Puzzle 2 Puzzle 3 Puzzle 4 Puzzle 5 /10 /10 /10 /10 /10
MATH-65 Puzzle Collection 6 Nov 8 :pm-:pm Name:... 3 :pm Wumaier :pm Njus 5 :pm Wumaier 6 :pm Njus 7 :pm Wumaier 8 :pm Njus This puzzle collection is closed book and closed notes. NO calculators are allowed
More informationSign the pledge. On my honor, I have neither given nor received unauthorized aid on this Exam : 11. a b c d e. 1. a b c d e. 2.
Math 258 Name: Final Exam Instructor: May 7, 2 Section: Calculators are NOT allowed. Do not remove this answer page you will return the whole exam. You will be allowed 2 hours to do the test. You may leave
More informationFirst Order Differential Equations f ( x,
Chapter d dx First Order Differential Equations f ( x, ).1 Linear Equations; Method of Integrating Factors Usuall the general first order linear equations has the form p( t ) g ( t ) (1) where pt () and
More information2tdt 1 y = t2 + C y = which implies C = 1 and the solution is y = 1
Lectures - Week 11 General First Order ODEs & Numerical Methods for IVPs In general, nonlinear problems are much more difficult to solve than linear ones. Unfortunately many phenomena exhibit nonlinear
More informationCalculus and Parametric Equations
Calculus and Parametric Equations MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction Given a pair a parametric equations x = f (t) y = g(t) for a t b we know how
More informationMA108 ODE: Picard s Theorem
MA18 ODE: Picard s Theorem Preeti Raman IIT Bombay MA18 Existence and Uniqueness The IVP s that we have considered usually have unique solutions. This need not always be the case. MA18 Example Example:
More informationMath 307 Lecture 6. Differences Between Linear and Nonlinear Equations. W.R. Casper. Department of Mathematics University of Washington
Math 307 Lecture 6 Differences Between Linear and Nonlinear Equations W.R. Casper Department of Mathematics University of Washington January 22, 2016 Today! Last time: Modeling first-order equations This
More informationExam II Review: Selected Solutions and Answers
November 9, 2011 Exam II Review: Selected Solutions and Answers NOTE: For additional worked problems see last year s review sheet and answers, the notes from class, and your text. Answers to problems from
More informationOn linear and non-linear equations.(sect. 2.4).
On linear and non-linear equations.sect. 2.4). Review: Linear differential equations. Non-linear differential equations. Properties of solutions to non-linear ODE. The Bernoulli equation. Review: Linear
More information= 2e t e 2t + ( e 2t )e 3t = 2e t e t = e t. Math 20D Final Review
Math D Final Review. Solve the differential equation in two ways, first using variation of parameters and then using undetermined coefficients: Corresponding homogenous equation: with characteristic equation
More information2.1 Differential Equations and Solutions. Blerina Xhabli
2.1 Math 3331 Differential Equations 2.1 Differential Equations and Solutions Blerina Xhabli Department of Mathematics, University of Houston blerina@math.uh.edu math.uh.edu/ blerina/teaching.html Blerina
More informationThe Fundamental Theorem of Calculus: Suppose f continuous on [a, b]. 1.) If G(x) = x. f(t)dt = F (b) F (a) where F is any antiderivative
1 Calulus pre-requisites you must know. Derivative = slope of tangent line = rate. Integral = area between curve and x-axis (where area can be negative). The Fundamental Theorem of Calculus: Suppose f
More informationSummer 2017 Session 1 Math 2410Q (Section 10) Elementary Differential Equations M-Th 4:45pm-7:00pm
Summer 2017 Session 1 Math 2410Q (Section 10) Elementary Differential Equations M-Th 4:45pm-7:00pm Instructor: Dr. Angelynn Alvarez E-mail: angelynn.alvarez@uconn.edu Office: MONT 305 Office Hours: MTuTh
More informationNumerical method for approximating the solution of an IVP. Euler Algorithm (the simplest approximation method)
Section 2.7 Euler s Method (Computer Approximation) Key Terms/ Ideas: Numerical method for approximating the solution of an IVP Linear Approximation; Tangent Line Euler Algorithm (the simplest approximation
More informationSection 3.6 The chain rule 1 Lecture. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I
Section 3.6 The chain rule 1 Lecture College of Science MATHS 101: Calculus I (University of Bahrain) Logarithmic Differentiation 1 / 23 Motivation Goal: We want to derive rules to find the derivative
More informationFirst Order ODEs (cont). Modeling with First Order ODEs
First Order ODEs (cont). Modeling with First Order ODEs September 11 15, 2017 Bernoulli s ODEs Yuliya Gorb Definition A first order ODE is called a Bernoulli s equation iff it is written in the form y
More informationFall 2016 Math 2B Suggested Homework Problems Solutions
Fall 016 Math B Suggested Homework Problems Solutions Antiderivatives Exercise : For all x ], + [, the most general antiderivative of f is given by : ( x ( x F(x = + x + C = 1 x x + x + C. Exercise 4 :
More informationChapter 1: Introduction
Chapter 1: Introduction Definition: A differential equation is an equation involving the derivative of a function. If the function depends on a single variable, then only ordinary derivatives appear and
More informationMATH 308 Differential Equations
MATH 308 Differential Equations Summer, 2014, SET 5 JoungDong Kim Set 5: Section 3.1, 3.2 Chapter 3. Second Order Linear Equations. Section 3.1 Homogeneous Equations with Constant Coefficients. In this
More information4. Higher Order Linear DEs
4. Higher Order Linear DEs Department of Mathematics & Statistics ASU Outline of Chapter 4 1 General Theory of nth Order Linear Equations 2 Homogeneous Equations with Constant Coecients 3 The Method of
More informationAgenda Sections 2.4, 2.5
Agenda Sections 2.4, 2.5 Reminders Read 3.1, 3.2 Do problems for 2.4, 2.5 Homework 1 due Friday Midterm Exam I on 1/23 Lab on Friday (Shapiro 2054) Office hours Tues, Thurs 3-4:30 pm (5852 East Hall) Theorem:
More informationNumerical method for approximating the solution of an IVP. Euler Algorithm (the simplest approximation method)
Section 2.7 Euler s Method (Computer Approximation) Key Terms/ Ideas: Numerical method for approximating the solution of an IVP Linear Approximation; Tangent Line Euler Algorithm (the simplest approximation
More informationPolytechnic Institute of NYU MA 2132 Final Practice Answers Fall 2012
Polytechnic Institute of NYU MA Final Practice Answers Fall Studying from past or sample exams is NOT recommended. If you do, it should be only AFTER you know how to do all of the homework and worksheet
More informationMAT 275 Laboratory 4 MATLAB solvers for First-Order IVP
MAT 275 Laboratory 4 MATLAB solvers for First-Order IVP In this laboratory session we will learn how to. Use MATLAB solvers for solving scalar IVP 2. Use MATLAB solvers for solving higher order ODEs and
More informationMath53: Ordinary Differential Equations Autumn 2004
Math53: Ordinary Differential Equations Autumn 2004 Unit 2 Summary Second- and Higher-Order Ordinary Differential Equations Extremely Important: Euler s formula Very Important: finding solutions to linear
More informationMath 23 Practice Quiz 2018 Spring
1. Write a few examples of (a) a homogeneous linear differential equation (b) a non-homogeneous linear differential equation (c) a linear and a non-linear differential equation. 2. Calculate f (t). Your
More informationHomework 9 - Solutions. Math 2177, Lecturer: Alena Erchenko
Homework 9 - Solutions Math 2177, Lecturer: Alena Erchenko 1. Classify the following differential equations (order, determine if it is linear or nonlinear, if it is linear, then determine if it is homogeneous
More informationHomework Solutions: , plus Substitutions
Homework Solutions: 2.-2.2, plus Substitutions Section 2. I have not included any drawings/direction fields. We can see them using Maple or by hand, so we ll be focusing on getting the analytic solutions
More informationMaximum and Minimum Values section 4.1
Maximum and Minimum Values section 4.1 Definition. Consider a function f on its domain D. (i) We say that f has absolute maximum at a point x 0 D if for all x D we have f(x) f(x 0 ). (ii) We say that f
More informationSolutions to Homework 3
Solutions to Homework 3 Section 3.4, Repeated Roots; Reduction of Order Q 1). Find the general solution to 2y + y = 0. Answer: The charactertic equation : r 2 2r + 1 = 0, solving it we get r = 1 as a repeated
More informationMAT 275 Laboratory 4 MATLAB solvers for First-Order IVP
MAT 75 Laboratory 4 MATLAB solvers for First-Order IVP In this laboratory session we will learn how to. Use MATLAB solvers for solving scalar IVP. Use MATLAB solvers for solving higher order ODEs and systems
More informationFirst Order Differential Equations
First Order Differential Equations 1 Finding Solutions 1.1 Linear Equations + p(t)y = g(t), y() = y. First Step: Compute the Integrating Factor. We want the integrating factor to satisfy µ (t) = p(t)µ(t).
More informationUS01CMTH02 UNIT-3. exists, then it is called the partial derivative of f with respect to y at (a, b) and is denoted by f. f(a, b + b) f(a, b) lim
Study material of BSc(Semester - I US01CMTH02 (Partial Derivatives Prepared by Nilesh Y Patel Head,Mathematics Department,VPand RPTPScience College 1 Partial derivatives US01CMTH02 UNIT-3 The concept of
More informationLecture 19: Solving linear ODEs + separable techniques for nonlinear ODE s
Lecture 19: Solving linear ODEs + separable techniques for nonlinear ODE s Geoffrey Cowles Department of Fisheries Oceanography School for Marine Science and Technology University of Massachusetts-Dartmouth
More informationOrdinary Differential Equation Theory
Part I Ordinary Differential Equation Theory 1 Introductory Theory An n th order ODE for y = y(t) has the form Usually it can be written F (t, y, y,.., y (n) ) = y (n) = f(t, y, y,.., y (n 1) ) (Implicit
More informationExistence Theory: Green s Functions
Chapter 5 Existence Theory: Green s Functions In this chapter we describe a method for constructing a Green s Function The method outlined is formal (not rigorous) When we find a solution to a PDE by constructing
More informationNonhomogeneous Equations and Variation of Parameters
Nonhomogeneous Equations Variation of Parameters June 17, 2016 1 Nonhomogeneous Equations 1.1 Review of First Order Equations If we look at a first order homogeneous constant coefficient ordinary differential
More informationEx. 1. Find the general solution for each of the following differential equations:
MATH 261.007 Instr. K. Ciesielski Spring 2010 NAME (print): SAMPLE TEST # 2 Solve the following exercises. Show your work. (No credit will be given for an answer with no supporting work shown.) Ex. 1.
More informationHomework 2 Solutions Math 307 Summer 17
Homework 2 Solutions Math 307 Summer 17 July 8, 2017 Section 2.3 Problem 4. A tank with capacity of 500 gallons originally contains 200 gallons of water with 100 pounds of salt in solution. Water containing
More informationMathematics Engineering Calculus III Fall 13 Test #1
Mathematics 2153-02 Engineering Calculus III Fall 13 Test #1 Instructor: Dr. Alexandra Shlapentokh (1) Which of the following statements is always true? (a) If x = f(t), y = g(t) and f (1) = 0, then dy/dx(1)
More informationDifferential Equations
This document was written and copyrighted by Paul Dawkins. Use of this document and its online version is governed by the Terms and Conditions of Use located at. The online version of this document is
More informationMath 308 Week 8 Solutions
Math 38 Week 8 Solutions There is a solution manual to Chapter 4 online: www.pearsoncustom.com/tamu math/. This online solutions manual contains solutions to some of the suggested problems. Here are solutions
More informationChain Rule. MATH 311, Calculus III. J. Robert Buchanan. Spring Department of Mathematics
3.33pt Chain Rule MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Spring 2019 Single Variable Chain Rule Suppose y = g(x) and z = f (y) then dz dx = d (f (g(x))) dx = f (g(x))g (x)
More informationIntroduction to Initial Value Problems
Chapter 2 Introduction to Initial Value Problems The purpose of this chapter is to study the simplest numerical methods for approximating the solution to a first order initial value problem (IVP). Because
More information6. Linear Differential Equations of the Second Order
September 26, 2012 6-1 6. Linear Differential Equations of the Second Order A differential equation of the form L(y) = g is called linear if L is a linear operator and g = g(t) is continuous. The most
More informationMath 132 Lab 3: Differential Equations
Math 132 Lab 3: Differential Equations Instructions. Follow the directions in each part of the lab. The lab report is due Monday, April 19. You need only hand in these pages. Answer each lab question in
More informationWork sheet / Things to know. Chapter 3
MATH 251 Work sheet / Things to know 1. Second order linear differential equation Standard form: Chapter 3 What makes it homogeneous? We will, for the most part, work with equations with constant coefficients
More informationSection , #5. Let Q be the amount of salt in oz in the tank. The scenario can be modeled by a differential equation.
Section.3.5.3, #5. Let Q be the amount of salt in oz in the tank. The scenario can be modeled by a differential equation dq = 1 4 (1 + sin(t) ) + Q, Q(0) = 50. (1) 100 (a) The differential equation given
More informationDON T PANIC! If you get stuck, take a deep breath and go on to the next question. Come back to the question you left if you have time at the end.
Math 307A, Midterm 1 Spring 2013 Name: Instructions. DON T PANIC! If you get stuck, take a deep breath and go on to the next question. Come back to the question you left if you have time at the end. There
More informationPartial proof: y = ϕ 1 (t) is a solution to y + p(t)y = 0 implies. Thus y = cϕ 1 (t) is a solution to y + p(t)y = 0 since
Existence and Uniqueness for LINEAR DEs. Homogeneous: y (n) + p 1 (t)y (n 1) +...p n 1 (t)y + p n (t)y = 0 Non-homogeneous: g(t) 0 y (n) + p 1 (t)y (n 1) +...p n 1 (t)y + p n (t)y = g(t) 1st order LINEAR
More informationA BRIEF INTRODUCTION INTO SOLVING DIFFERENTIAL EQUATIONS
MATTHIAS GERDTS A BRIEF INTRODUCTION INTO SOLVING DIFFERENTIAL EQUATIONS Universität der Bundeswehr München Addresse des Autors: Matthias Gerdts Institut für Mathematik und Rechneranwendung Universität
More informationwe get y 2 5y = x + e x + C: From the initial condition y(0) = 1, we get 1 5 = 0+1+C; so that C = 5. Completing the square to solve y 2 5y = x + e x 5
Math 24 Final Exam Solution 17 December 1999 1. Find the general solution to the differential equation ty 0 +2y = sin t. Solution: Rewriting the equation in the form (for t 6= 0),we find that y 0 + 2 t
More informationSolutions to Homework 5
Solutions to Homework 5 1. Let z = f(x, y) be a twice continuously differentiable function of x and y. Let x = r cos θ and y = r sin θ be the equations which transform polar coordinates into rectangular
More informationFind the Fourier series of the odd-periodic extension of the function f (x) = 1 for x ( 1, 0). Solution: The Fourier series is.
Review for Final Exam. Monday /09, :45-:45pm in CC-403. Exam is cumulative, -4 problems. 5 grading attempts per problem. Problems similar to homeworks. Integration and LT tables provided. No notes, no
More informationTrue or False. Circle T if the statement is always true; otherwise circle F. for all angles θ. T F. 1 sin θ
Math 90 Practice Midterm III Solutions Ch. 8-0 (Ebersole), 3.3-3.8 (Stewart) DISCLAIMER. This collection of practice problems is not guaranteed to be identical, in length or content, to the actual exam.
More informationSolving systems of ODEs with Matlab
Solving systems of ODEs with Matlab James K. Peterson Department of Biological Sciences and Department of Mathematical Sciences Clemson University October 20, 2013 Outline 1 Systems of ODEs 2 Setting Up
More informationVector Functions & Space Curves MATH 2110Q
Vector Functions & Space Curves Vector Functions & Space Curves Vector Functions Definition A vector function or vector-valued function is a function that takes real numbers as inputs and gives vectors
More informationLecture 2. Introduction to Differential Equations. Roman Kitsela. October 1, Roman Kitsela Lecture 2 October 1, / 25
Lecture 2 Introduction to Differential Equations Roman Kitsela October 1, 2018 Roman Kitsela Lecture 2 October 1, 2018 1 / 25 Quick announcements URL for the class website: http://www.math.ucsd.edu/~rkitsela/20d/
More informationES.1803 Topic 7 Notes Jeremy Orloff. 7 Solving linear DEs; Method of undetermined coefficients
ES.1803 Topic 7 Notes Jeremy Orloff 7 Solving linear DEs; Method of undetermined coefficients 7.1 Goals 1. Be able to solve a linear differential equation by superpositioning a particular solution with
More informationName: Solutions Final Exam
Instructions. Answer each of the questions on your own paper. Put your name on each page of your paper. Be sure to show your work so that partial credit can be adequately assessed. Credit will not be given
More information1 Solution to Homework 4
Solution to Homework Section. 5. The characteristic equation is r r + = (r )(r ) = 0 r = or r =. y(t) = c e t + c e t y = c e t + c e t. y(0) =, y (0) = c + c =, c + c = c =, c =. To find the maximum value
More informationQuiz 4A Solutions. Math 150 (62493) Spring Name: Instructor: C. Panza
Math 150 (62493) Spring 2019 Quiz 4A Solutions Instructor: C. Panza Quiz 4A Solutions: (20 points) Neatly show your work in the space provided, clearly mark and label your answers. Show proper equality,
More information